L茅vy-like movement patterns of metastatic cancer cells revealed in microfabricated systems and implicated in vivo AbstractMetastatic cancer cells differ from their non-metastatic counterparts not only in terms of molecular composition and genetics, but also by the very strategy they employ for locomotion. Here, we analyzed large-scale statistics for cells migrating on linear microtracks to show that metastatic cancer cells follow a qualitatively different movement strategy than their non-invasive counterparts. The trajectories of metastatic cells display clusters of small steps that are interspersed with long 鈥渇lights鈥? Such movements are characterized by heavy-tailed, truncated power law distributions of persistence times and are consistent with the L茅vy walks that are also often employed by animal predators searching for scarce prey or food sources. In contrast, non-metastatic cancerous cells perform simple diffusive movements. These findings are supported by preliminary experiments with cancer cells migrating away from primary tumors in vivo. The use of chemical inhibitors targeting actin-binding proteins allows for 鈥渞eprogramming鈥?the L茅vy walks into either diffusive or ballistic movements. IntroductionThe motility of mammalian cells has been studied for decades1,2, and trajectories of cell movements have been quantified in various ways. Early models of cell motility were founded on the classic Langevin equation and described the movements of adherent cells3,4,5 (for description of smaller, faster, and weakly-adherent immune cells, see ref. 6,7) as an Ornstein鈥揢hlenbeck (OU) process,8 such that the cell鈥檚 mean square displacement,鈥?lt;鈥?i>x(t)2鈥?gt;鈥? is expressed as 2nD (t 鈥?P (1 鈥?exp (-t/P)), where n is the dimensionality of the system, D is the diffusion coefficient, and P is the so-called persistence time. This model predicts Gaussian distribution of velocities that are exponentially correlated in time, leading to directional persistence on short time scales (t鈥?i> 鈥?i>P)9. On long time scales (t 禄 P), the model is reduced to a random walk and predicts uniform distribution of turning angles. This formalism has been successfully applied to describe the motions鈥攃oined persistent random walks (PRWs)鈥攐f fibroblasts, lung epithelial cells, and microvessel endothelial cells3,4,5. In addition, due to apparent directional persistence, animal as well as cellular movements have frequently been described as 鈥渃orrelated random walks鈥?(CRWs)7,10,11. In CRWs1, the step sizes/times are drawn from the Gaussian or other exponentially decaying distribution, and the direction of the preceding step influences the direction of the next step. Overall, both PRW and CRW models predict that cell movements are ballistic (i.e., persistent in direction or鈥?lt;鈥?i>x2鈥?i> 鈥墌 t伪 with 伪鈥?鈥?) at short times and diffusive (伪鈥?鈥?) at long time scales. L茅vy walks鈥攔ecently detected in T cells6鈥攁re different because they are superdiffusive12 (i.e.,鈥?lt;鈥?i>x2鈥?i> 鈥墌 t伪 with 1鈥?lt;鈥?i>伪鈥?lt;鈥?) at all times and are composed of sequences of many short steps interspersed with longer \"flights\". This pattern is conserved across all scales, in effect giving rise to fractal patterns with no characteristic scale13. Mathematically, L茅vy walks14,15 are characterized by non-Gaussian, heavy-tailed, power law distribution of persistence times/step sizes, such that P(t)鈥?i>~鈥?i>t -渭, where t is persistence time/step size or time/distance it takes to move one step between the turns and 渭 is power law (L茅vy) exponent with 1鈥?lt;鈥?i>渭鈥?lt;鈥?. In the absence of a characteristic scale, the overall length of a L茅vy walk is determined by the longest step and the step-length variance grows over time, though it remains finite even when unbounded by biological and environmental considerations. Most biological systems are bounded/limited (e.g., cell trajectories are limited by cell cycle and environmental conditions), resulting in truncation of the power law tail14,16,17,18, which introduces characteristic scale to the movement pattern. However, variability around the characteristic scale is very large and self-similar, which is in sharp contrast to other finite-scale movement patterns. L茅vy-like movement patterns have been observed in movements of a number of multicellular animals17,18,19,20,21,22,23,24,25 and humans26,27, found in trace fossil trails17, and recently also in T cells searching for parasite-infected cells6, swarming bacteria28,29, and even molecular motors within cells16. The observation of L茅vy walks has been attributed to the execution of an optimal search strategy for sparsely and randomly distributed resources/target sites30,31,32,33, though this interpretation has not been generally accepted13,34.With the general applicability of each of these models still being debated,9,35 one outstanding and important question concerns the movement patterns of non-metastatic versus metastatic cancer cells. Although the latter are known to have higher migration velocity and, in some cases, increased directional persistence36,37,38,39, it remains unclear whether increased metastatic potential is reflected by a qualitative change in the overall strategy of cell locomotion.In this work, we address this question with a material system of micropatterned lines10,40 on which the cells perform one-dimensional motions which (i) have been shown41 to mimic the motility of cells migrating in 3D better than commonly used planar substrates and (ii) are observed in metastasizing tumor cells in vivo, attaching preferentially to and moving along linear fibers (e.g., collagen fibers36) or along preexisting linear perimuscular or perineural 鈥渕icrochannels鈥?sup>42,43. Importantly, the 鈥渕icrotracks鈥?we use enable unambiguous determination of persistence times鈥攁s the times that cells move 鈥渢o the left鈥?or 鈥渢o the right鈥?before reverting their direction of motion44鈥攁nd collection of large numbers of data points such that even the low-probability events are captured. Statistical analysis of such data then reveals that while non-metastatic cancerous cells are simple diffusive movers, the metastatic ones not only move in a superdiffusive fashion (which has been shown before45,46 but not in the context of metastatic cells), but also display movement patterns consistent with L茅vy walks1,47. Significantly, we also detect L茅vy walks of metastatic cells migrating in preliminary in vivo studies, in which we used state-of-the-art intravital multiphoton microscopy to resolve trajectories of individual cancer cells migrating away from animal-implanted tumors. While the generality of the observed trends certainly merits additional studies (especially in vivo), the results we describe pose some intriguing questions as to why the invasive cancerous cells have developed a movement strategy that is frequently observed in multicellular animal predators and hunters17,18,19,20,21,22,23,24,25 (as well as killer/effector T cells6 searching for rare target cells) and, as mentioned above, is thought to correspond to an optimal search strategy for sparsely and randomly distributed resources/target sites30,31,32,33 (but see also,34). In this context, toward the end of the paper, we describe experiments in which RNA interference and chemical inhibition of actin-binding proteins can change the L茅vy walking phenotype of metastatic 鈥減redators鈥?into either unidirectional, ballistic motions, or into diffusive migrations characterizing benign or non-invasive cancerous cells.ResultsL茅vy walks of metastatic cancer cells revealed on linear microtracksMost of the in vitro experiments were performed on linear microtracks etched in gold-on-glass substrates using the so-called Wet Etching technique40,48,49 (Fig.聽1a; for all fabrication details, see Supplementary聽Methods/Supplementary Note聽1). The gold regions were protected with self-assembled monolayers, SAMs50,51, of oligo (ethylene glycol)-terminated alkyl thiols (HS(CH2)11(OCH2CH2)6OH; EG6 (ProChimia Surfaces, Gdansk, Poland: www.prochimia.com) known to prevent cell adhesion. The unprotected glass lines were covered with either Laminin (Sigma-Aldrich, cat. # L2020) or Laminin 5 (LN 5; extracted from 804G cells in crude form, as previously described52)鈥攖hese extracellular matrix substrates were chosen because of their motility-promoting characteristics (vs. more adhesive fibronectin) and because they are routinely used as physiologically relevant substrates of choice for the respective cell lines.Fig. 1Trajectories of cancer cell motions on linear microtracks. a Scheme of substrate fabrication using the Wet Etching technique40, 48, 49. b Optical micrographs showing a cell migrating on a microtrack (scale bar鈥?鈥?0鈥壜祄). The cell moves along the etched, optically transparent microtrack, and not across cell-adhesion-resistant dark regions. The definition of step persistence length/time is the distance/time the cell travels in one direction before it reverts the direction of motion. The arrows labeled L1, L2, and L3 indicate three consecutive 鈥渟teps鈥?of the cell (here, to the right, to the left, and to the right again). c A representative trajectory of a metastatic cell comprised of聽clusters of 鈥渟mall鈥?steps (shown in gray) interspersed with 鈥渓arge鈥?steps (color denotes elapsed time and each long step is in different color) is characteristic of a L茅vy walk (see also Supplementary Figure聽2 for long-term trajectories). Scale bar is 100鈥壩糾 for L茅vy trajectory and 20鈥壩糾 for the inset. This can be contrasted with a trajectory of a non-metastatic cell exhibiting diffusive motion (all steps are small and shown in gray, scale bar is 20鈥壩糾). Note that while cell motions in experiments are in 1D (along microtracks), the vertical axis in the trajectories shown here corresponds to time (from top to bottom). Total length of each trajectory is 960鈥塵in with each point 3鈥塵in apart. See also Supplementary Movies聽1鈥?a data-track=\"click\" data-track-label=\"link\" data-track-action=\"supplementary material anchor\" href=\"/articles/s41467-018-06563-w#MOESM8\">6. The distinction between 鈥渟mall鈥?and 鈥渓arge鈥?steps is best appreciated by viewing聽long-term Supplementary Movies聽13鈥?a data-track=\"click\" data-track-label=\"link\" data-track-action=\"supplementary material anchor\" href=\"/articles/s41467-018-06563-w#MOESM17\">15Full size imageWhen the cells were applied (at plating density of ~10,000鈥塩ells/cm2) onto microstructured substrates presenting arrays of 20-渭m-wide linear tracks, they localized exclusively onto these tracks, spread, and, to a good approximation, displayed one-dimensional motions (Fig.聽1b). We compared and contrasted motions of six types of cells from three cancers (Fig.聽2; Supplementary Figure聽1): non-metastatic PC-3 and metastatic PC-3M53 prostate cancer cells; non-metastatic MCF-7 and metastatic MDA-MB-23138 breast cancer cells; and non-metastatic B16-F0 and metastatic B16-F154 mouse melanoma cells. Regarding the cell line choices, we note that for B16 and PC lines, cells are termed metastatic versus non-metastatic based on, respectively, their low and high metastatic potentials53,54. For breast cancer lines, the MCF-7 cell line retains several characteristics of differentiated mammary epithelium and represents a聽poorly invasive luminal subtype of breast cancer, whereas the MDA-MB-231 line represents a聽highly invasive basal subtype of breast cancer55.Fig. 2Superdiffusive and L茅vy walks of metastatic cancer cells on linear microtracks. a Typical trajectories/displacement versus time of highly metastatic cells (here for MDA-MB-231) feature characteristic small steps interspersed with unidirectional, long excursions. b In contrast, trajectories of non-metastatic cells (here for MCF-7) are more random/鈥漥iggly鈥? Ten representative trajectories per cell type are shown. The starting points for trajectories are randomly positioned along the y axis (鈥淒istance鈥? for clarity. See also Supplementary Movies聽1鈥?a data-track=\"click\" data-track-label=\"link\" data-track-action=\"supplementary material anchor\" href=\"/articles/s41467-018-06563-w#MOESM8\">6 and 13鈥?a data-track=\"click\" data-track-label=\"link\" data-track-action=\"supplementary material anchor\" href=\"/articles/s41467-018-06563-w#MOESM17\">15 and Supplementary Figure聽1 for trajectories for PC-3, PC-3M, B16-F0, and B16-F1 cells and Supplementary Figure聽2 for long-term trajectories. c Differences in the two modes of motility are quantified in the log鈥搇og plots of the cells鈥?mean square displacement (in 渭m2) versus time, (leftlangle {x^2} rightrangle propto t^alpha). The values of 伪 close to unity (PC-3: 伪鈥?i>=鈥?.04, 95% confidence interval鈥壜扁€?.03; MCF-7: 伪鈥?鈥?.96鈥壜扁€?.04; B16F0: 伪鈥?鈥?.05鈥壜扁€?.02) indicate diffusive walks of non-metastatic cells. Metastatic cells are superdiffusive (PC-3M: 伪鈥?鈥?.58鈥壜扁€?.02; MD-MB-231: 伪鈥?鈥?.54鈥壜扁€?.01; B16F1: 伪鈥?鈥?.52鈥壜扁€?.02). d鈥?b>f The cumulative frequency distributions, CFDs, of persistence times (t) for all types of cells studied on microtracks. Markers are experimental statistics: magneta triangles for PC-3, red crosses for PC-3M, blue crosses for MDA-MB-231, orange rectangles for MCF-7, green circles for B16-F0, and black circles for B16-F1. Solid lines are theoretical truncated power law fits. Statistical analysis of cancer cell movements on 1D microtracks is shown in Table聽1Full size imageUsing an automated image acquisition and analysis protocol developed in-house, we were able to monitor cell motions for up to 16鈥塰 and collect robust statistics with n鈥?鈥?7鈥?9 different cells and ~5120鈥?0,800 time points per every cell type studied (see Table聽1 legend). Only tracks housing one cell (~60% of all tracks at the plating density used) were analyzed to eliminate any potential artifacts due to cell鈥揷ell collisions (see Supplementary Note聽1 for comment on cell collisions). The typical trajectories of cells on the microtracks shown in Fig.聽1c, Fig.聽2a, b, and Supplementary Figure聽1 are characteristic of, respectively, metastatic and non-metastatic cells (see Supplementary Movies聽1 vs.聽2, 3 vs.聽4 and 5 vs.聽6; also Supplementary Movies聽13鈥?a data-track=\"click\" data-track-label=\"link\" data-track-action=\"supplementary material anchor\" href=\"/articles/s41467-018-06563-w#MOESM17\">15).Table 1 Statistical analysis of cancer cell movements on 1D microtracksFull size tableTo quantify these cell trajectories and the dynamics of cell motions, we first plotted cells鈥?mean square displacements, (leftlangle {x^2} rightrangle = leftlangle {left[ {x(t + t_0) - x(t_0)} right]^2} rightrangle), versus time, t, where (leftlangle {...} rightrangle) denotes a combined average over聽all starting times t0 and cell paths (see refs. 12,45). When plotted on a log鈥搇og scale, these dependencies give straight lines corresponding to (leftlangle {x^2} rightrangle = t^alpha) scaling, with the slope of the lines being the exponent 伪. From statistical physics, it is well known12 that 伪鈥?鈥? indicates diffusive motion, values 1鈥?lt;鈥?i>伪鈥?i> 2 correspond to superdiffusion, and 伪鈥?鈥? is characteristic of ballistic motion (e.g., unidirectional motion without turns). As evidenced by the plots in Fig.聽2c, all non-metastatic cells we studied move diffusively (伪鈥?i>~鈥?) while their highly metastatic variants move in a superdiffusive fashion (伪鈥?i>~鈥?.5鈥?.6). These characteristics are conserved over the entire time domain and not only short term (as CRW or PRWs).Long-term superdiffusive behavior is sometimes found in systems performing L茅vy walks,47 which are a sub-class of random walks and are characterized by periods of small steps interspersed with long but infrequent unidirectional excursions.聽Typical cell trajectories such as those shown in Fig.聽1c and Fig.聽2a (see also Supplementary Figure聽1鈥? and Supplementary Movies聽1, 3, 5, and 13鈥?a data-track=\"click\" data-track-label=\"link\" data-track-action=\"supplementary material anchor\" href=\"/articles/s41467-018-06563-w#MOESM17\">15) suggest that the metastatic MDA-MB-231, PC-3M, and B16-F1 cells might indeed be performing L茅vy walks. This hypothesis can be mathematically verified by the analysis of cells\' cumulative frequency distributions, CFDs, of persistence times (i.e., the persistence time is defined as the time that cell moves persistently in a given direction12). When such distributions are plotted on a log鈥搇og scale, it is evident that the CFDs corresponding to metastatic cells are more heavy-tailed than CFDs corresponding to non-metastatic ones聽(Fig. 2 d鈥揻). However, extreme care must be taken when assigning such trends to a specific functional form, especially given recent findings56,57 that the use of inaccurate statistical methods has led to an incorrect assignment of the search/movement patterns as L茅vy flights in a number of studies. Accordingly, we followed the rigorous procedure developed by Edwards et al.57,58 to test power law distributions using the likelihood and Akaike weights (which measure an appropriateness of a given fit57,58) and tested multiple alternative competing models (see also聽Supplementary Notes聽1 and 2, and Supplementary Tables聽1, 2) as detailed by Clauset et al56. In addition to power law and exponential models, we also considered heavy-tailed log-normal (observed in motions of T cells within lymph nodes7), heavy-tailed stretched exponential56, as well as a truncated power law59 (in which power law tail of L茅vy walk is truncated).In particular, the truncated power law was considered because data collection time was limited for both short and long times. The short time limitation at ~3鈥?鈥塵in was due to relatively large cell sizes and small speeds (respectively, ~20鈥?0鈥壩糾 and ~0.25鈥?鈥壩糾/min for metastatic cells vs.鈥?i>~鈥?鈥壩糾 and鈥?i>~鈥?鈥?2鈥壩糾/min for T cells studied in ref. 6), making it difficult聽to (1) distinguish their translocation from just the membrane dynamics, and (2) limit the photo-damage imposed to cells by laser/light exposure. The long time limitation was due to the cells dividing (e.g., ~30鈥塰 for MDA-MB-231). It is therefore impossible to obtain single-cell data spanning the desirable two decades of persistence times and enhancing statistics of low-probability, long cellular excursions in the power law鈥檚 tail60. Instead,聽we (akin to others9,35,46) have analyzed persistence-time distributions over the populations of single cells (see also Supplementary Figure聽4 for single-cell analysis). This being said, to capture as many long excursions as possible, in some experiments we managed to record individual cellular trajectories for up to 40鈥塰 (if cells divided during observation time, they were tracked only up to division event) and total displacements up to 1000鈥壩糾 (see Supplementary Figures聽2, 5, Supplementary Table聽3, and Supplementary Movies聽13鈥?a data-track=\"click\" data-track-label=\"link\" data-track-action=\"supplementary material anchor\" href=\"/articles/s41467-018-06563-w#MOESM17\">15).The results summarized in Fig.聽2d鈥揻 and Table聽1 demonstrate that only the metastatic cells (PC-3M, MDA-MB-231, B16-F1) exhibit log鈥搇og distributions of persistence times that can be assigned as L茅vy walks. While the cumulative frequency distribution, CFD, plots might 鈥渁ppear鈥?to fit a simple power law ((P(t) = (mu - 1)a^{mu - 1}t^{ - mu },t ge a), where ((mu - 1)a^{mu - 1}) is the normalization constant and parameter a determines the time at which a power law fit is appropriate; a values are listed in Table聽1), detailed maximum likelihood and Akaike-weight analyses evidence that a more appropriate fit is a truncated power law, (Pleft( t right) = left( {frac{{mu - 1}}{{a^{1 - mu } - b^{1 - mu }}}} right)t^{ - mu }), where b is an upper truncation parameter corresponding to the maximal point in each dataset. Importantly, the values of exponents 碌 in these fits are below 3, corresponding to L茅vy walks. Moreover, such analyses confirm that the truncated power law fit is superior to all alternative models considered (Fig.聽2d鈥揻, Table聽1, see also Supplementary Figure聽5 for CFDs and model fits; Supplementary Figure聽6 for evaluation of log-normal distribution; Supplementary Tables聽1 and 2 and Supplementary聽Methods for all additional details on model fitting and comparisons). All data sets for metastatic cells pass the聽goodness-of-fit test performed according to the method detailed in ref. 56 (Supplementary Note聽3 and Supplementary Figure聽7). In this context, it is important to note that truncated power law (with 碌 ~1鈥?) has become increasingly favored (over pure power law) as a more relevant model for describing bounded/limited biological systems and鈥攕imilar to a pure power law鈥攊s considered L茅vy walk14,16,17,18. We also note that (i) 鈥渨alks鈥?rather than 鈥渇lights鈥?sup>15 terminology is appropriate since, as we verified, our migrating cells exhibit strong correlation between persistence times and persistence lengths (i.e., step sizes); and (ii) observing L茅vy-like patterns in a simple 1D system in the absence of chemoattractant gradients suggests that they may be an inherent characteristic of metastatic cells.Regarding non-metastatic cells, their CFDs also fit best to a truncated power law, but the exponents 渭 in these fits are greater than 3 (Fig.聽2d鈥揻 and Table聽1), which is in line with the purely diffusive34,47 nature of these cells鈥?motions.L茅vy walks of metastatic cancer cells in live tumorsWhile the microtracks offer a convenient platform for collecting large numbers of motility data, these ex vivo results are not necessarily indicative of in vivo motility patterns, which may be influenced by local chemotactic gradients and/or cell interactions with the tissue microenvironment. Extension to in vivo, however, is technically challenging36,61 and further complicated in our case where resolving individual cells over long (hours) times is necessary for the collection of statistics ample enough to construct reliable probability distributions. Also, it should be remembered that in vivo analyses cannot offer complete correspondence with all the cell types studied on microtracks and are limited to tumors (i) lying within the penetration depth of the existing high-resolution microscopy modalities (max. 600鈥壩糾 in mouse dermis), and (ii) for which appropriate animal protocols have been validated and approved. In our study, the additional requirement to compare metastatic versus non-metastatic cells of the same origin limited the available choices to a pair of non-metastatic B16-F0 and metastatic B16-F10 mouse melanoma cells. The trajectories of these cells invading the mouse dermis were recorded using a dorsal skin-fold chamber model43. Using combined near-infrared and infrared multiphoton microscopy43, second-harmonic generation (SHG) allowed for the reconstruction of fibrillar collagen and myofibers, while fluorescence enabled visualization of blood vessels and tracking of moving cell nuclei (Fig.聽3).Fig. 3Structure of the linear invasion strands in vivo. a Live B16-F0 (non-metastatic) and B16-F10 (metastatic) tumors in mouse skin imaged with epifluorescence microscopy (cell nuclei marked with Histone-2B/mCherry, green). Images are representative of least six tumors from at least three independent mice per group. Insets show finger-like invasion strands moving away from the main tumor mass. Scale bar is 100鈥壩糾. b Enlarged images corresponding to the tips of the invasion strands. Few (~2鈥?) B16-F10 cells, so-called tip cells, detached from the invasion strands and exhibited trajectories that over limited observation time appeared ballistic (see also Fig.聽4b, Zone 5). These tip cells were observed only in the very outer zone 5 which was not included in L茅vy walk analysis because of small number of cells in this zone. Scale bar is 100鈥壩糾. Quantification of (c) tumor growth (quantification based on tumor volume and expressed as an increase over volume measured on day 1; time, days) and (d) invasion strand length on day 6. Invasion strands were statistically longer in metastatic tumors (average ~400鈥壩糾 in B16-F10 vs. ~100鈥壩糾 in B16-F0, red horizontal lines). Error bars in c聽are standard deviations based on鈥夆墺鈥?0 invasion strands from at least six tumors of three individual mice. e The widths of invasion strands near the base (i.e., beginning of strand at tumor鈥檚 edge), center between base and tip, and the tip region (recorded on day 6; error bars give standard deviations of 鈮? invasion strands from at least four tumors of three individual mice). f Individualized (with separation above two nuclear diameters) versus compact cell positioning in leading tips of invasion strandsFull size imageOn one hand, both B16-F0 and B16-F10 cells predominantly formed invasion strands consisting of up to 100 individual cells which jointly infiltrated the dermis by following linear tissue interfaces provided by parallel perimuscular and perineural microtracks, blood vessels, or collagen bundles (Fig.聽3). On the other hand, the path organization and kinetics of cell motions were markedly different for the non-metastatic (Fig.聽4a and Supplementary Movie聽7) and metastatic cells (Fig.聽4b and Supplementary Movie聽8). To quantify these motions, we divided the entire tumor/strand domain into 150-渭m wide zones with zone 1 near the border of the tumor mass and incremental outward numbering along the invasion zone (see Fig.聽4a, b and Table聽2). The path organizations were found to be strongly zone-dependent. At the tumor margin (zone 1), where cells are 鈥渃rowded鈥?and frequent cell鈥揷ell collisions occur, the motions of both B16-F0 and B16-F10 cells are 鈥渏iggly鈥?and purely diffusive, as evidenced by the 鈥渄iffusion exponent鈥?伪鈥?i>~鈥? (Fig.聽4c). Beyond the tumor border, within zone 2, the cells experience more frequent collisions from one side (zone 1) and are effectively 鈥減ushed away鈥?from the tumor exhibiting slightly differing degrees of superdiffusive motions (伪鈥?i>~鈥?.38 for B16-F0 and 伪鈥?i>~鈥?.63 for B16-F10). The differences between the inherent characteristics of cell motions鈥攊.e., not dominated by crowding effects/cell鈥揷ell collisions鈥攎anifest themselves fully in zones 3 and 4, with non-metastatic cells reverting to diffusive motions (伪鈥?i>~鈥?) while metastatic cells remaining evidently superdiffusive, with 伪鈥?i>~鈥?.47鈥?.87. These characteristics were conserved over the entire time domain (i.e., not only short term as in correlated/persistent random walks) and were statistically significant as evidenced by the 95% confidence intervals for 伪 listed in Table聽2.Fig. 4Migration of non-metastatic and metastatic cells from live tumors in mouse skin. a, b Cell movements within invasion strands formed by up to 100 tumor cells and relation to the guidance structures of the tumor microenvironment, observed by using high-resolution multiphoton microscopy (see Supplementary Movies聽7 and 8 for trajectories). The snapshots from these movies are shown for (a) B16-F0 and (b) B16-F10 tumors. AlexaFluor 750-labeled dextran was used to visualize blood vessels (red), Histone-2B/mCherry marks the nuclei of cells (green), and fibrillar collagen was revealed by SHG (second-harmonic generation; blue). Dashed yellow lines divide the invasion strands into five zones, each 150-渭m wide. Scale bar is 150鈥壩糾. c The values of the 鈥渄iffusion exponents鈥?伪 indicate that both B16-F0 and B16-F10 cells are diffusive at the tumor margin (zone 1, 伪~1) and superdiffusive (zone 2, 伪~1.4鈥?.6) with entering the invasion zone. Away from the tumor (zones 3鈥?), however, the metastatic cells remain superdiffusive while the non-metastatic ones show diffusive behavior. Error bars correspond to 95% confidence intervals. d The cumulative frequency distributions, CFDs, of persistence times are truncated power law with 渭鈥?i> 鈥?i>3 for diffusive B16-F0 (blue crosses, Zones 1鈥?) and truncated power law with 渭~2.36 for B16-F10 (red circles, Zones 2鈥?) performing L茅vy walks. Markers are experimental statistics for persistence times. Solid lines are truncated power law fits. Statistical analysis of cancer cell movements in vivo is shown in Table聽2Full size imageTable 2 Statistical analysis of cancer cell movements in vivoFull size tableMost importantly, the cumulative frequency distributions, CFDs, of persistence times shown in Fig.聽4d reveal that non-metastatic B16-F0聽cells move diffusively (truncated power law with 渭鈥?i> 鈥?i>3), whereas the metastatic B16-F10聽cells have a CFD characteristic of a L茅vy walk (truncated power law with 2鈥?lt;鈥?i>渭鈥?i> 鈥?i>3). For the B16-F0聽cells, the Akaike weights quantifying the appropriateness of the fits show that even though truncated power law fit is preferred over pure power law or exponential fits, 渭鈥?i> 鈥?i>3 indicates diffusive motion for all zones. For B16-F10聽cells, truncated power law fit is also appropriate in all zones with Akaike weights close to 1, but the value of exponent 2鈥?lt;鈥?i>渭鈥?i> 鈥?i>3 for zones 2鈥? indicates L茅vy-like motion (Table聽2). When analyzing all data for B16-F10 from zones 2鈥? together, 碌鈥?i>~鈥?.36鈥壜?0.13 with Akaike weights for truncated power law close to 1. Overall, these in vivo studies resemble the results obtained for cell鈥檚 on microtracks both in terms of the diffusive-vs-L茅vy dichotomy and the linearity of cell trajectories (here, along tissue micro-channels rather than microtracks). We note that additional analysis of cells migrating through 3D collagen gels (to probe the effects of 1D vs. 3D microenvironment) are also consistent with diffusive-vs-L茅vy motions for normal versus cancer cells (see Supplementary Note聽4 and Supplementary Figures聽8, 9).鈥淩eprogramming鈥?L茅vy walks into other types of motionsAssuming that the L茅vy-type walks might be an untoward characteristic of 鈥減redatory鈥?metastatic cells, we next focused on the question whether these walks could somehow be altered鈥攊n particular鈥攃ould they be reverted into diffusive motions characterizing the non-invasive cancer cells?To this end, we inhibited鈥攅ither chemically or by using specific siRNAs鈥攕elected proteins known to be involved in polymerization of actin filaments and their organization into higher-order structures and known to drive extension of cellular protrusions62. Results summarized in Fig.聽5 and Table聽3 (see also Supplementary Movies聽9鈥?a data-track=\"click\" data-track-label=\"link\" data-track-action=\"supplementary material anchor\" href=\"/articles/s41467-018-06563-w#MOESM14\">12 and 16鈥?a data-track=\"click\" data-track-label=\"link\" data-track-action=\"supplementary material anchor\" href=\"/articles/s41467-018-06563-w#MOESM20\">18) for the metastatic MDA-MB-231 cells moving on the microtracks reveal that inhibiting Rac1 (with chemical inhibitor NSC23766) or knocking down of Cofilin-1 (involved in actin filament depolymerization), Profilin-1 (regulating actin polymerization), or Dia-1 (involved in polymerization of unbranched actin filaments) increased 渭 exponents slightly, but not as much as to fully eliminate L茅vy-like motions, (Table聽3, see also Supplementary Figures聽10鈥?a data-track=\"click\" data-track-label=\"link\" data-track-action=\"supplementary material anchor\" href=\"/articles/s41467-018-06563-w#MOESM1\">13). In contrast, inhibition of the Arp2/3 (but not of upstream GTPase Rac1, Fig.聽5b and Supplementary Movie聽11)鈥?involved in the polymerization of actin into dendritic/branched networks at cell鈥檚 鈥渇ront鈥濃€攂y a small-molecule inhibitor CK666 not only slowed the metastatic cells down, from 0.99鈥壩糾/min to 0.57鈥壩糾/min, but changed the nature of their motions from L茅vy-like to diffusive (Fig.聽5d, Supplementary Figure聽14, Supplementary Movies聽9, 17, and Supplementary Table聽4 for the聽summary of all speed data). When Myosin II鈥攖ypically involved in forming contractile actin networks and bundles and causing concurrent cell rear retraction62鈥攚as inhibited by blebbistatin at intermediate inhibitor concentrations (see聽 Fig.聽5聽legend), the cell speed changed only slightly from 0.99鈥壩糾/min to 1.13鈥壩糾/min, but caused cells to move ballistically (i.e., steadily in one direction; note that in the context of this work, 鈥渂allistic鈥?corresponds to motions for which persistence times are longer than total observation time and for which 伪 ~2; Fig.聽5c and Supplementary Movie聽10). In addition, pronounced changes in the motility patterns could also be achieved by inhibition of several proteins simultaneously. A case in point here is the reversal of L茅vy to diffusive walks by the simultaneous inhibition of Myosin II with 10鈥壩糓 of blebbistatin and of Rac-1 with 100鈥壩糓 of NSC23766 (Supplementary Figures聽11 and 13, Supplementary Movies聽12 and 18)鈥攚hat is interesting about this result is that the inhibition of each of the proteins separately (cf. Fig.聽5b, c) does not lead to diffusive walking, implying a synergistic/cooperative action of the two proteins in controlling the L茅vy-walk movement pattern.Fig. 5Altering the motility strategy of metastatic cancer cells. Examples of typical cell trajectories: L茅vy walking (control MDA-MB-231 cells, a or with Rac1 inhibited, b), ballistic (Myosin II inhibited, c) diffusive (Arp2/3 inhibited, d). Line width鈥?鈥?0鈥壩糾, same for all three images. In displacement plots, ten representative trajectories for each treatment are shown. Quantification of motility characteristics (exponents 渭, 伪 along with the鈥壜扁€?5% confidence intervals) for MDA-MD-231 cells moving on microtracks and having individual actin-binding proteins inhibited is summarized in Table聽3. e Log鈥搇og plots of the cells鈥?mean square displacement versus time, (leftlangle {x^2} rightrangle = t^alpha). The slopes correspond to exponents 伪; note that inhibition of Arp2/3 with CK666 results in diffusive motion, 伪鈥?i>~鈥?, while inhibition of Myosin II results in ballistic motion, 伪~2. f The corresponding cumulative frequency distributions, CFDs, of persistence times. Markers are experimental statistics for persistence times. Solid lines are truncated power law fits with respective 渭 values shown in Table聽3. For the chemical inhibitors data shown corresponds to: 40鈥壩糓 CK666 (for Arp2/3, yellow crosses), 100鈥壩糓 NSC23755 (Rac1, green triangles), 10鈥壩糓 Blebbistatin (Myosin II, red rectangles), and control (black circles). The additional results for all drug and siRNA concentrations tested are shown in Supplementary Figures聽10鈥?a data-track=\"click\" data-track-label=\"link\" data-track-action=\"supplementary material anchor\" href=\"/articles/s41467-018-06563-w#MOESM1\">13. See Supplementary Movies聽9鈥?a data-track=\"click\" data-track-label=\"link\" data-track-action=\"supplementary material anchor\" href=\"/articles/s41467-018-06563-w#MOESM14\">12 and 16鈥?a data-track=\"click\" data-track-label=\"link\" data-track-action=\"supplementary material anchor\" href=\"/articles/s41467-018-06563-w#MOESM20\">18Full size imageTable 3 Statistical analysis of metastatic cancer cell movements upon inhibition of actin regulatorsFull size tableThe role of protrusion-retraction synchronization in L茅vy walksOur previous morphodynamic profiling analyses have shown that in metastatic cells, the protrusions and retractions are highly 鈥渟ynchronized鈥?both in space and in time; in contrast, protrusions and retractions formed by non-metastatic cells are not 鈥渟ynchronized鈥?sup>63. In order to test if such protrusion-retraction synchronization can give rise to truncated L茅vy walk/power law step-size distributions, we developed a simple model described in detail in Supplementary Note聽5, Supplementary Figures聽15 and 16. Briefly, we considered a scenario in which the probabilities of taking left-right steps depend on prior history (non-Markovian process). We showed that if consecutive steps slightly favor moves in the 鈥渟ame鈥?direction 鈥?e.g., as observed in the microscale dynamics of cell membrane where persistence of membrane protrusion typically depends on levels of actin regulators, such as Rac1 and Arp2/3, activation of the聽positive feedback loops, precise spatiotemporal regulation of Rho family GTPases, and coupling of protrusion to substrate adhesion2,62鈥攖hen the overall distribution of persistence length can follow truncated power law (Supplementary Figures聽15 and 16). The general conclusion of this modeling effort is, therefore, that synchronization (or desynchronization) of front-back protrusions/retractions may determine the overall motility pattern. The model indicates that cells in which front and back dynamics are synchronized are expected to perform truncated L茅vy walks (as in our experiments with metastatic cells) whereas lack of synchronization should translate into diffusive motion (as in non-metastatic cells and metastatic cells treated with inhibitors). Experimentally, we observe some signatures of 鈥渄esynchronization鈥?of front-rear protrusion-retractions in MDA-MB-231 cells treated with inhibitors (or their combination) that revert L茅vy to diffusive motion (see Supplementary Figure聽14 for details and Supplementary Movies聽16鈥?a data-track=\"click\" data-track-label=\"link\" data-track-action=\"supplementary material anchor\" href=\"/articles/s41467-018-06563-w#MOESM20\">18). Based on these results, we propose a hypothesis that L茅vy-like movement pattern in metastatic cancer cells on 1D microtracks results from the balance/competition of persistent protrusion鈥攙ia synchronized front-rear motions dependent on Arp2/3鈥攁nd an effective myosin II-dependent mechanism for switching the direction of cell motion.DiscussionTaken together, the above results establish the presence of L茅vy-like movement patterns in a range of metastatic cancer cells, and also suggest means of changing this motility phenotype. Metastatic cells not only move 鈥渇aster and more persistently鈥?sup>10 than their non-metastatic counterparts, but while doing so, also display a fundamentally distinct path structure. This 鈥減redatory鈥?navigation strategy is independent of external chemotactic signals as gradients of various chemokines, EGF, or VEGF are all absent in the microtrack experiments. This work contributes to the growing body of knowledge indicating that L茅vy-like movement patterns are present and fundamentally important not only for humans26,27 and multicellular animals17,18,19,20,21,22,23,24,25, but also on cellular6,28,29 and subcellular levels16.L茅vy walk movement patterns are generally different from and should not be confused with classical models of cell motions such as persistent and simple correlated random walks (PRW/CRW reviewed by1). The latter are generally characterized by ballistic (i.e., persistent in direction) motions at short time and diffusive at long time scales. In contrast, L茅vy walks are superdiffusive over long times thus allowing the searcher to move farther away from the starting point in a聽shorter time than CRW/PRW strategy would. This being said, it should be noted that recent theoretical analyses and experimental evidence from multicellular animal movements indicate that related movement patterns, specifically multi-phasic walks (e.g., composite CRWs and composite Brownian walks), in which the mover switches between two or more kinds of simple walk patterns (mathematically characterized by step-sizes from two or more exponential distributions which can sum up to a power law distribution60) have been suggested 鈥渢o have parameters that are fine-tuned to [optimal] L茅vy walk鈥?sup>17,59. While bi-phasic (or bi-modal) walks have realistic physical correlates (such as run phases and reorientation/wait phases)64, it seems difficult to find physical justification for fitting/using more complex multi-exponential distributions to describe step-size distributions, such as the ones we observe here for metastatic cells. In addition, discrete nature of cell motility data we collected precluded reliable fitting of bi-exponential distributions (see Supplementary Note聽2 for more information).While our observations of L茅vy walk motility pattern for metastatic cells moving through very simple experimental environments in the absence of chemotactic gradients suggest that L茅vy walk is an intrinsic property of metastatic cancer cells, current work does not exclude the possibility that constrained complex geometries, such as aligned collagen fibers and linear micro-tunnels/tracks, contribute to the emergence of the聽L茅vy walk pattern in complex in vivo environments. In this context, it is interesting that recent theoretical studies have shown that linear constraints, such as one-dimensional micro-channels, provide one of the simplest systems for the emergence of optimal L茅vy walks23,32. For example, in the system described by Reynolds et al.23, a L茅vy-like displacement patterns (so-called Weierstrassian L茅vy walk23,29) can emerge as a result of the moving object bouncing chaotically off the walls of the channel. While intriguing, such minimalistic mechanical explanation might not be at work in our 1D microtrack system simply because there are no 鈥渨alls鈥? and cellular motions are determined mostly by cell鈥搒ubstrate adhesive interactions and are gently guided by adhesive micropatterns. At the same time, cellular interactions with complex geometric/mechanical constraints could play role in wider micro-channels/tracks, such as in PDMS micro-channels used by others65 and in vivo settings36,43.To the best of our knowledge, the work presented here is the first demonstration of L茅vy-like movement patterns in adherent mammalian cells, as well as in the context of metastatic cancer. Regarding the latter aspect, we ponder whether adopting the聽L茅vy-like modality of migration might endow metastatic cells with a聽successful strategy for dispersing and searching for suitable loci where to seed new metastases. The prevailing theory (though disputed by others,34) stipulates that L茅vy walk searches are considered to be optimal within a聽narrow range of circumstances30,31,32,33. Specifically, non-destructive L茅vy walk searches with 渭~ 2 were considered optimal provided that targets are randomly distributed and scarce and searcher does not have prior knowledge about the target locations30,31,32. However, more recent theoretical work has shown that L茅vy walk searches (again, with 渭~ 2) are optimal under much broader environmental conditions than previously thought and that L茅vy searchers experience fewer long periods of starvation (compared with exponential, composite Brownian and ballistic searchers)33,66. Furthermore, the so-called adaptive L茅vy searching (where searcher has some knowledge of the target distribution and, upon target detection, responds by switching from extensive L茅vy searching to intensive Brownian searching) has been shown to outperform adaptive ballistic and composite Brownian searches67. Interestingly, the聽theoretical 2D path structure analysis33,34 highlights a remarkable difference in movement patterns between L茅vy (with 渭鈥?/i>~ 2.5, close to values observed in some of our experiments) and exponential searchers. The former takes many small steps in a focused area resulting in thorough focused exploration (high oversampling) until one large step takes it to another area (similar to 1D path structure of our metastatic cells, see Supplementary Movies聽13鈥?a data-track=\"click\" data-track-label=\"link\" data-track-action=\"supplementary material anchor\" href=\"/articles/s41467-018-06563-w#MOESM17\">15). In contrast, an exponential searcher diffuses more gradually and covers the area more evenly without a聽thorough exploration of any particular spots33. This otherwise sub-optimal L茅vy searcher (渭~ 2.5) performs extremely well at intermediate levels of prey/target abundance. In the context of metastasis, focused local exploration characteristic of L茅vy walk (with 渭~ 2.5) may be useful for finding 鈥渉ot spots鈥?of defective basement membranes68 followed by ballistic motions along linear collagen fibers and/or micro-channels toward another set of basement membranes before entering the bloodstream. Interestingly, risk of predation鈥攃ondition relevant to metastatic cancer cells which during dissemination in vivo are constantly targeted by immune cells鈥攁lters optimal search strategy such that searchers executing L茅vy walk with 2鈥?lt;鈥?i>渭鈥?lt;鈥? have higher fitness than those with 渭鈥夆墹鈥?69. Taken together, our work suggests that L茅vy-like movements represent an emergent strategy of how highly motile metastatic cells move within complex microenvironments, and because such movement patterns could be advantageous, they may be maintained and/or fine-tuned by selection pressures.Naturally, the present results should be generalized to as many types of cells/cancers and conditions (for example, micro-channels) as possible, though especially the in vivo studies are currently limited by the inability to visualize tumors lying deeper into the animal and to obtain long-enough trajectories. In addition, we do not yet fully understand the origin of L茅vy walks at the level of cytoskeletal dynamics. While we have developed a simple model (see Supplementary Note聽5 and Supplementary Figures聽15, 16) that could ascribe these walks by the synchronization of the聽cell鈥檚 鈥渇ront鈥?protrusion (driven by Arp2/3-mediated nucleation of actin filaments) and 鈥渂ack鈥?retraction (enabled by actomyosin contraction), such assumptions need further scrutiny and experimental validation. We see further effort in this area justified by the hope that by eliminating the ability of metastatic cells to execute 鈥減redatory鈥?L茅vy walks through the human body, it might be possible to decrease the ability of these cells to seed metastatic cancers.MethodsIn vivo observation of cell trajectoriesThe movements of stable Histone2B/EGF or Histone2B/mCherry expressing non-metastatic B16-F0 and metastatic B16-F10 mouse melanoma cells were observed in live tumors established in mouse dermis by using the dorsal skin-fold chamber model43. Dorsal skin-fold chambers were transplanted on a skin flap of C57/B16 J mice (Charles River) and B16-F0 or B16-F10 Histone2B-EGFP/mCherry cells were implanted by injection of 5鈥壝椻€?04鈥?鈥壝椻€?05 cells into the dermis adjacent to the deep dermal vascular plexus. For visualization of blood vessels, AlexaFluor 750-labeled 70鈥塳D dextran was injected intravenously (2鈥塵g/mouse). A customized multiphoton microscope (TriMScope-II, LaVisionBioTec) setup was used which allowed for simultaneous second-harmonic generation (SHG) to reconstruct tissue interfaces, and fluorescence imaging to track blood vessels and the movements of B16-F0 and B16-F10 cells stably expressing Histone-2B/GFP in live mouse dermis. Time-lapse acquisition was done 3鈥? days after tumor cell implantation for 1鈥?鈥塰 at 5 or 10鈥塵in intervals.All animal experiments were approved by the ethical committee on animal experiments and performed in the Central Animal Laboratory of the Radboud University, Nijmegen, in accordance with the Dutch Animal Experimentation Act and the European FELASA protocol (www.felasa.eu/guidelines.php).Tracking cell movements on 1D microtrack substratesTracks for cell locomotion were microetched in gold-on-glass substrates using the so-called Wet Etching40,48,49 technique (see Fig.聽1a) where unetched gold regions were protected against cell adhesion with oligo(ethylene glycol)-terminated alkyl thiols, (HS(CH2)11(OCH2CH2)6OH (ProChimia Surfaces, Gdansk, Poland: www.prochimia.com). PC-3, PC3-M, MCF-7, MDA-MB-231, B16-F0 and B16-F1 cells were cultured according to the American Type Culture Collection (ATCC) protocol or as described previously40,48,49. Linear tracks were coated with Laminin (Sigma-Aldrich, cat. #L2020) or Laminin 5 (LN 5; extracted from 804G cells in crude form as previously described52). Cell migration was monitored on inverted microscope (TMD, Nikon) equipped with phase-contrast optics (10鈥壝椻€? 0.25 NA objectives) and CCD camera (Sensys, Photometrics, Tucson, AZ). Image acquisition was driven by Metamorph software (Universal Imaging Corp., Worchester, PA). Time-lapse videos were collected over 16鈥塰 in 3-min intervals. Cell positions were assigned by their center-of-mass coordinates.Data analysisHeavy-tailed models and fitting procedures: The equations used in the comparisons of the heavy-tailed models can be found in Supplementary Table聽1. Power law and exponential distribution parameters were estimated using analytical expression for maximum likelihood estimator, as discussed elsewhere56,57. Parameters for truncated power law, log-normal, and stretched exponential distributions were found using numerical maximum likelihood estimation and verified by visual inspection of likelihood maps. The appearance of largely negative 渭 values in case of log-normal model is demonstrated in Supplementary Figure聽6. Effectively, the log-normal curve is being stretched to better fit power law-like distribution of the data, suggesting that log-normal is not the optimal model for the data. When using only positive 渭, the resulting log-normal likelihood and, therefore, wAIC are worse. The resulting parameters are shown in the Supplementary Table聽2. Parameter a is a lower cutoff parameter, the choice of which is described below, and b is an upper cutoff parameter used in truncated power law estimation, and corresponds to the maximal point in each dataset.Lower cutoff estimation: To estimate the choice of the lower cutoff parameter a, we used a technique based on reweighted Kolmogorov鈥揝mirnov (rKS) statistic, described in detail in ref. 56. In brief, the rKS is calculated for all a values taken from the set of unique values of persistence times in each dataset. Then, a is chosen where rKS is minimal and the number of remaining data points after thresholding by a is not less than 50% of the original dataset. In cases of truncated and regular power law, exponent values in Table聽1 are shown for lower cutoffs determined separately using rKS, whereas wAIC for all distributions are calculated using cutoff values for the truncated power law.For all other experimental details, please see Supplementary聽Methods/聽Supplementary Notes聽1鈥? online. Data and code used to generate results in the current study are available from the corresponding authors upon reasonable request. References1.Codling, E. A., Plank, M. J. Benhamou, S. Random walk models in biology. J. R. Soc. Interface 5, 813鈥?34 (2008).Article聽Google Scholar聽 2.Lauffenburger, D. A. Horwitz, A. F. Cell migration: a physically integrated molecular process. Cell 84, 359鈥?69 (1996).CAS聽 Article聽Google Scholar聽 3.Dunn, G. A. Brown, A. F. A unified approach to analysing cell motility. J. Cell. Sci. Suppl. 8, 81鈥?02 (1987).CAS聽 Article聽Google Scholar聽 4.Gail, M. H. Boone, C. W. The locomotion of mouse fibroblasts in tissue culture. Biophys. J. 10, 980鈥?93 (1970).ADS聽 CAS聽 Article聽Google Scholar聽 5.Stokes, C. L., Lauffenburger, D. A. Williams, S. K. Migration of individual microvessel endothelial cells: stochastic model and parameter measurement. J. Cell. Sci. 99, 419鈥?30 (1991).PubMed聽Google Scholar聽 6.Harris, T. H. et al. Generalized L茅vy walks and the role of chemokines in migration of effector CD8(+) T cells. Nature 486, 545鈥?48 (2012).ADS聽 CAS聽 Article聽Google Scholar聽 7.Fricke, G. M., Letendre, K. A., Moses, M. E. Cannon, J. L. Persistence and adaptation in immunity: T Cells balance the extent and thoroughness of search. PLoS. Comput. Biol. 12, e1004818 (2016).ADS聽 Article聽Google Scholar聽 8.Uhlenbeck, G. E. Ornstein, L. S. On the theory of the Brownian motion. Phys. Rev. 36, 823鈥?41 (1930).ADS聽 CAS聽 Article聽Google Scholar聽 9.Wu, P. H., Giri, A., Sun, S. X. Wirtz, D. Three-dimensional cell migration does not follow a random walk. Proc. Natl Acad. Sci. USA 111, 3949鈥?954 (2014).ADS聽 CAS聽 Article聽Google Scholar聽 10.Maiuri, P. et al. The first world cell race. Curr. Biol. 22, R673鈥揜675 (2012).CAS聽 Article聽Google Scholar聽 11.Li, L., Norrelykke, S. F. Cox, E. C. Persistent cell motion in the absence of external signals: a search strategy for eukaryotic cells. PLoS. ONE. 3, e2093 (2008).ADS聽 Article聽Google Scholar聽 12.Metzler, R. Klafter, J. The random walk鈥檚 guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1鈥?7 (2000).ADS聽 MathSciNet聽 CAS聽 Article聽Google Scholar聽 13.Reynolds, A. Liberating L茅vy walk research from the shackles of optimal foraging. Phys. Life Rev. 14, 59鈥?3 (2015).ADS聽 Article聽Google Scholar聽 14.Mantegna, R. N. Stanley, H. E. Stochastic-process with ultraslow convergence to a gaussian - the truncated L茅vy Flight. Phys. Rev. Lett. 73, 2946鈥?949 (1994).ADS聽 MathSciNet聽 CAS聽 Article聽Google Scholar聽 15.Klafter, J., Blumen, A. Shlesinger, M. F. Stochastic pathway to anomalous diffusion. Phys. Rev. A. 35, 3081鈥?085 (1987).ADS聽 MathSciNet聽 CAS聽 Article聽Google Scholar聽 16.Chen, K. J., Wang, B. Granick, S. Memoryless self-reinforcing directionality in endosomal active transport within living cells. Nat. Mater. 14, 589鈥?93 (2015).ADS聽 CAS聽 Article聽Google Scholar聽 17.Sims, D. W. et al. Hierarchical random walks in trace fossils and the origin of optimal search behavior. Proc. Natl Acad. Sci. USA 111, 11073鈥?1078 (2014).ADS聽 CAS聽 Article聽Google Scholar聽 18.Humphries, N. E., Weimerskirch, H., Queiroz, N., Southall, E. J. Sims, D. W. Foraging success of biological L茅vy flights recorded in situ. Proc. Natl Acad. Sci. USA 109, 7169鈥?174 (2012).ADS聽 CAS聽 Article聽Google Scholar聽 19.Viswanathan, G. M. et al. L茅vy flight search patterns of wandering albatrosses. Nature 381, 413鈥?15 (1996).ADS聽 CAS聽 Article聽Google Scholar聽 20.Sims, D. W. et al. Scaling laws of marine predator search behaviour. Nature 451, 1098鈥?102 (2008).ADS聽 CAS聽 Article聽Google Scholar聽 21.Humphries, N. E. et al. Environmental context explains L茅vy and Brownian movement patterns of marine predators. Nature 465, 1066鈥?069 (2010).ADS聽 CAS聽 Article聽Google Scholar聽 22.Kolzsch, A. et al. Experimental evidence for inherent L茅vy search behaviour in foraging animals. Proc. Biol. Sci. 282, 20150424 (2015).Article聽Google Scholar聽 23.Reynolds, A., Santini, G., Chelazzi, G. Focardi, S. The Weierstrassian movement patterns of snails. R. Soc. Open Sci. 4, 160941 (2017).MathSciNet聽 Article聽Google Scholar聽 24.Focardi, S., Montanaro, P. Pecchioli, E. Adaptive L茅vy walks in foraging fallow deer. PLoS. ONE. 4, e6587 (2009).ADS聽 Article聽Google Scholar聽 25.Reynolds, A. M. et al. Displaced honey bees perform optimal scale-free search flights. Ecology 88, 1955鈥?961 (2007).Article聽Google Scholar聽 26.Raichlen, D. A. et al. Evidence of L茅vy walk foraging patterns in human hunter-gatherers. Proc. Natl Acad. Sci. USA 111, 728鈥?33 (2014).ADS聽 CAS聽 Article聽Google Scholar聽 27.Reynolds, A., Ceccon, E., Baldauf, C., Karina Medeiros, T. Miramontes, O. Levy foraging patterns of rural humans. PLoS. ONE. 13, e0199099 (2018).Article聽Google Scholar聽 28.Ariel, G. et al. Swarming bacteria migrate by L茅vy walk. Nat. Commun. 6, 8396 (2015).CAS聽 Article聽Google Scholar聽 29.Ariel, G., Be鈥檈r, A. Reynolds, A. Chaotic model for L茅vy walks in swarming bacteria. Phys. Rev. Lett. 118, 228102 (2017).ADS聽 Article聽Google Scholar聽 30.Viswanathan, G. M. et al. Optimizing the success of random searches. Nature 401, 911鈥?14 (1999).ADS聽 CAS聽 Article聽Google Scholar聽 31.Raposo, E. P. et al. Dynamical robustness of L茅vy search strategies. Phys. Rev. Lett. 91, 240601 (2003).ADS聽 CAS聽 Article聽Google Scholar聽 32.Viswanathan, G. M., Raposo, E. P. da Luz, M. G. E. L茅vy flights and superdiffusion in the context of biological encounters and random searches. Phys. Life Rev. 5, 133鈥?50 (2008).ADS聽 Article聽Google Scholar聽 33.Humphries, N. E. Sims, D. W. Optimal foraging strategies: L茅vy walks balance searching and patch exploitation under a very broad range of conditions. J. Theor. Biol. 358, 179鈥?93 (2014).Article聽Google Scholar聽 34.James, A., Plank, M. J. Edwards, A. M. Assessing L茅vy walks as models of animal foraging. J. R. Soc. Interface 8, 1233鈥?247 (2011).Article聽Google Scholar聽 35.Selmeczi, D., Mosler, S., Hagedorn, P. H., Larsen, N. B. Flyvbjerg, H. Cell motility as persistent random motion: theories from experiments. Biophys. J. 89, 912鈥?31 (2005).CAS聽 Article聽Google Scholar聽 36.Condeelis, J. Segall, J. E. Intravital imaging of cell movement in tumours. Nat. Rev. Cancer 3, 921鈥?30 (2003).CAS聽 Article聽Google Scholar聽 37.Partin, A. W., Schoeniger, J. S., Mohler, J. L. Coffey, D. S. Fourier analysis of cell motility: correlation of motility with metastatic potential. Proc. Natl Acad. Sci. USA 86, 1254鈥?258 (1989).ADS聽 CAS聽 Article聽Google Scholar聽 38.Sliva, D., Mason, R., Xiao, H. English, D. Enhancement of the migration of metastatic human breast cancer cells by phosphatidic acid. Biochem. Biophys. Res. Commun. 268, 471鈥?79 (2000).CAS聽 Article聽Google Scholar聽 39.Chicoine, M. R. Silbergeld, D. L. The in vitro motility of human gliomas increases with increasing grade of malignancy. Cancer 75, 2904鈥?909 (1995).CAS聽 Article聽Google Scholar聽 40.Kandere-Grzybowska, K. et al. Cell motility on micropatterned treadmills and tracks. Soft Matter 3, 672鈥?79 (2007).ADS聽 CAS聽 Article聽Google Scholar聽 41.Doyle, A. D., Wang, F. W., Matsumoto, K. Yamada, K. M. One-dimensional topography underlies three-dimensional fibrillar cell migration. J. Cell. Biol. 184, 481鈥?90 (2009).CAS聽 Article聽Google Scholar聽 42.Wolf, K. et al. Multi-step pericellular proteolysis controls the transition from individual to collective cancer cell invasion. Nat. Cell Biol. 9, 893鈥?04 (2007).CAS聽 Article聽Google Scholar聽 43.Weigelin, B., Bakker, G.-J. Friedl, P. Intravital third harmonic generation microscopy of collective melanoma cell invasion. Intravital 1, 32043 (2012).Article聽Google Scholar聽 44.Huda, S. et al. Microfabrication tools: microfabricated systems and assays for studying the cytoskeletal organization, micromechanics, and motility patterns of cancerous cells. Adv. Mat. Interfaces 1, 1400158 (2014).Article聽Google Scholar聽 45.Dieterich, P., Klages, R., Preuss, R. Schwab, A. Anomalous dynamics of cell migration. Proc. Natl Acad. Sci. USA 105, 459鈥?63 (2008).ADS聽 CAS聽 Article聽Google Scholar聽 46.Takagi, H., Sato, M. J., Yanagida, T. Ueda, M. Functional analysis of spontaneous cell movement under different physiological conditions. PLoS. ONE. 3, e2648 (2008).ADS聽 Article聽Google Scholar聽 47.Shlesinger, M. F., Zaslavsky, G. M. Klafter, J. Strange kinetics. Nature 363, 31鈥?7 (1993).ADS聽 CAS聽 Article聽Google Scholar聽 48.Kandere-Grzybowska, K., Campbell, C., Komarova, Y., Grzybowski, B. A. Borisy, G. G. Molecular dynamics imaging in micropatterned living cells. Nat. Methods 2, 739鈥?41 (2005).CAS聽 Article聽Google Scholar聽 49.Mahmud, G. et al. Directing cell motions on micropatterned ratchets. Nat. Phys. 5, 606鈥?12 (2009).CAS聽 Article聽Google Scholar聽 50.Love, J. C., Estroff, L. A., Kriebel, J. K., Nuzzo, R. G. Whitesides, G. M. Self-assembled monolayers of thiolates on metals as a form of nanotechnology. Chem. Rev. 105, 1103鈥?169 (2005).CAS聽 Article聽Google Scholar聽 51.Witt, D., Klajn, R., Barski, P. Grzybowski, B. A. Applications properties and synthesis of omega-functionalized n-alkanethiols and disulfides - the building blocks of self-assembled monolayers. Curr. Org. Chem. 8, 1763鈥?797 (2004).CAS聽 Article聽Google Scholar聽 52.Langhofer, M., Hopkinson, S. B. Jones, J. C. R. The matrix secreted by 804g cells contains laminin-related components that participate in hemidesmosome assembly in-vitro. J. Cell. Sci. 105, 753鈥?64 (1993).CAS聽 PubMed聽Google Scholar聽 53.Liu, Y. Q. et al. Prostate cancer chemoprevention agents exhibit selective activity against early stage prostate cancer cells. Prostate Cancer P. D. 4, 81鈥?1 (2001).CAS聽 Article聽Google Scholar聽 54.Fidler, I. J. Selection of successive tumor lines for metastasis. Nat.-New Biol. 242, 148鈥?49 (1973).CAS聽 Article聽Google Scholar聽 55.Neve, R. M. et al. A collection of breast cancer cell lines for the study of functionally distinct cancer subtypes. Cancer Cell. 10, 515鈥?27 (2006).CAS聽 Article聽Google Scholar聽 56.Clauset, A., Shalizi, C. R. Newman, M. E. J. Power-law distributions in empirical data. SIAM Rev. 51, 661鈥?03 (2009).ADS聽 MathSciNet聽 Article聽Google Scholar聽 57.Edwards, A. M. et al. Revisiting L茅vy flight search patterns of wandering albatrosses, bumblebees and deer. Nature 449, 1044鈥?048 (2007).ADS聽 CAS聽 Article聽Google Scholar聽 58.Edwards, A. M. Using likelihood to test for L茅vy flight search patterns and for general power-law distributions in nature. J. Anim. Ecol. 77, 1212鈥?222 (2008).Article聽Google Scholar聽 59.Jansen, V. A. A., Mashanova, A. Petrovskii, S. Comment on 鈥淟茅vy walks evolve through interaction between movement and environmental complexity鈥? Science 335, 918 (2012).ADS聽 CAS聽 Article聽Google Scholar聽 60.Stumpf, M. P. H. Porter, M. A. Critical truths about power laws. Science 335, 665鈥?66 (2012).ADS聽 MathSciNet聽 CAS聽 Article聽Google Scholar聽 61.Tozluoglu, M. et al. Matrix geometry determines optimal cancer cell migration strategy and modulates response to interventions. Nat. Cell Biol. 15, 751鈥?62 (2013).CAS聽 Article聽Google Scholar聽 62.Ridley, A. J. et al. Cell migration: Integrating signals from front to back. Science 302, 1704鈥?709 (2003).ADS聽 CAS聽 Article聽Google Scholar聽 63.Hermans, T. M. et al. Motility efficiency and spatiotemporal synchronization in non-metastatic vs. metastatic breast cancer cells. Integr. Biol. (Camb.) 5, 1464鈥?473 (2013).CAS聽 Article聽Google Scholar聽 64.Potdar, A. A., Jeon, J., Weaver, A. M., Quaranta, V. Cummings, P. T. Human mammary epithelial cells exhibit a bimodal correlated random walk pattern. PLoS. ONE. 5, e9636 (2010).ADS聽 Article聽Google Scholar聽 65.Irimia, D. Toner, M. Spontaneous migration of cancer cells under conditions of mechanical confinement. Integr. Biol. (Camb.) 1, 506鈥?12 (2009).CAS聽 Article聽Google Scholar聽 66.Wosniack, M. E., Santos, M. C., Raposo, E. P., Viswanathan, G. M. da Luz, M. G. E. The evolutionary origins of L茅vy walk foraging. PLoS. Comput. Biol. 13, e1005774 (2017).ADS聽 Article聽Google Scholar聽 67.Reynolds, A. M. Adaptive Levy walks can outperform composite Brownian walks in non-destructive random searching scenarios. Phys. A Stat. Mech. Appl. 388, 561鈥?64 (2009).Article聽Google Scholar聽 68.Clark, A. G. Vignjevic, D. M. Modes of cancer cell invasion and the role of the microenvironment. Curr. Opin. Cell Biol. 36, 13鈥?2 (2015).CAS聽 Article聽Google Scholar聽 69.Abe, M. S. Shimada, M. Levy Walks Suboptimal under Predation Risk. PLoS. Comput. Biol. 11, e1005601 (2015).Article聽Google Scholar聽 70.Bartumeus, F., Catalan, J., Fulco, U. L., Lyra, M. L. Viswanathan, G. M. Optimizing the encounter rate in biological interactions: L茅vy versus Brownian strategies. Phys. Rev. Lett. 88, 097901 (2002).ADS聽 CAS聽 Article聽Google Scholar聽 Download referencesAcknowledgementsThis work was supported by (1) National Institutes of Health (NIH, USA) Awards #1R21CA137707鈥?1, #5R21CA173347-01, #1R21CA173232 and #U54CA119341 to B.A.G.; (2) Non-Equilibrium Energy Research Center (NERC), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences (DE-SC0000989) to B.A.G.; (3) the Netherlands Science Organization (NWO-VICI 918.11.626) and the Cancer Genomics Center to P.F.; (4) the Netherlands Science Organization (NWO-VIDI 917.10.364) to K.W.; and (5) the Institute for Basic Science, Republic of Korea, (IBS-R020-D1) to B.A.G. We thank Dr. Yaroslav Sobolev for help with developing final version of the theoretical model of L茅vy walks described in Supplementary Note聽5.Author informationAffiliationsDepartment of Chemical and Biological Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL, 60208, USASabil Huda,聽Gary Wilk,聽Fateme S. Emami,聽Siowling Soh,聽Didzis Pilans,聽Amir Vahid聽 聽Monika MakurathDepartment of Cell Biology (283) RIMLS, Radboud University Medical Centre, Geert Grooteplein 28, 6525, GA, Nijmegen, The NetherlandsBettina Weigelin,聽Katarina Wolf聽 聽Peter FriedlDavid H. Koch Center for Applied Research of Genitourinary Cancers, Department of Genitourinary Medical Oncology, The University of Texas MD Anderson Cancer Center, Houston, TX, 77030, USABettina Weigelin聽 聽Peter FriedlInstitute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17/19, 60-179, Pozna艅, PolandKonstantin V. Tretiakov聽 聽Jakub W. NarojczykIBS Center for Soft and Living Matter, Ulsan National Institute of Science and Technology (UNIST), 50 UNIST-gil, Ulju-gun, 689鈥?98, South KoreaKonstantin Polev,聽Kristiana Kandere-Grzybowska聽 聽Bartosz A. GrzybowskiDepartment of Biomedical Engineering, School of Life Sciences, Ulsan National Institute of Science and Technology (UNIST), 50 UNIST-gil, Ulju-gun, 689-798, South KoreaKonstantin Polev聽 聽Kristiana Kandere-GrzybowskaCenter for General Education, Aichi Institute of Technology, 1247 Yachigusa Yakusacho, Toyota, 470-0392, JapanMasatomo IwasaFaculty of Physics and NanoBioMedicine Centre, Adam Mickiewicz University, Umultowska 85, 61-614, Pozna艅, PolandMichal BanaszakCancer Genomics Centre Netherlands (CG.nl), Utrecht, NetherlandsPeter FriedlThe Forsyth Institute, 245 First St., Cambridge, MA, 02142, USAGary G. BorisyDepartment of Chemistry, Ulsan National Institute of Science and Technology (UNIST), 50 UNIST-gil, Ulju-gun, 689-798, South KoreaBartosz A. GrzybowskiAuthorsSabil HudaView author publicationsYou can also search for this author in PubMed聽Google ScholarBettina WeigelinView author publicationsYou can also search for this author in PubMed聽Google ScholarKatarina WolfView author publicationsYou can also search for this author in PubMed聽Google ScholarKonstantin V. TretiakovView author publicationsYou can also search for this author in PubMed聽Google ScholarKonstantin PolevView author publicationsYou can also search for this author in PubMed聽Google ScholarGary WilkView author publicationsYou can also search for this author in PubMed聽Google ScholarMasatomo IwasaView author publicationsYou can also search for this author in PubMed聽Google ScholarFateme S. EmamiView author publicationsYou can also search for this author in PubMed聽Google ScholarJakub W. NarojczykView author publicationsYou can also search for this author in PubMed聽Google ScholarMichal BanaszakView author publicationsYou can also search for this author in PubMed聽Google ScholarSiowling SohView author publicationsYou can also search for this author in PubMed聽Google ScholarDidzis PilansView author publicationsYou can also search for this author in PubMed聽Google ScholarAmir VahidView author publicationsYou can also search for this author in PubMed聽Google ScholarMonika MakurathView author publicationsYou can also search for this author in PubMed聽Google ScholarPeter FriedlView author publicationsYou can also search for this author in PubMed聽Google ScholarGary G. BorisyView author publicationsYou can also search for this author in PubMed聽Google ScholarKristiana Kandere-GrzybowskaView author publicationsYou can also search for this author in PubMed聽Google ScholarBartosz A. GrzybowskiView author publicationsYou can also search for this author in PubMed聽Google ScholarContributionsS.H. performed microtrack experiments, analyzed the data, contributed to the biological interpretation of the results and made the figures; B.W. performed and analyzed in vivo experiments; K.W. performed and analyzed 3D collagen gel experiments; G.W., D.P., M.M. and K.K.G performed experiments; K.V.T., K.P., G.W. and M.I. wrote computer codes for data analysis, analyzed the results, and provided their theoretical interpretation; M.B. and S.S. helped with the theoretical analysis; F.S.E., A.V. and J.W.N. developed theoretical model of L茅vy walks described in the SI; G.G.B. and P.F. designed experiments and helped with biological interpretation of the results; K.K.G. and B.A.G. conceived and supervised the research, designed experiments, and wrote the paper. All authors read and corrected the paper.Corresponding authorsCorrespondence to Kristiana Kandere-Grzybowska or Bartosz A. Grzybowski.Ethics declarations Competing interests The authors declare no competing interests. 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Drohan, Luis Chiriboga, Jessica Llewellyn, Cheng Z. Liu, Young Nyun Park, Rebecca G. Wells Neil D. Theise Communications Biology (2021) Val茅rie C. Reijers, Koen Siteur, Selwyn Hoeks, Jim van Belzen, Annieke C. W. Borst, Jannes H. T. Heusinkveld, Laura L. Govers, Tjeerd J. Bouma, Leon P. M. Lamers, Johan van de Koppel Tjisse van der Heide Nature Communications (2019) CommentsBy submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. Sign up for the Nature Briefing newsletter 鈥?what matters in science, free to your inbox daily.